Determinants and inverses

Idea: One number that decides a lot

The determinant says whether a matrix can be undone and how much it stretches area or volume.

• Zero: the plane is squashed and there is no inverse
• Negative: the map also flips orientation
• It sits in the denominator of every inverse entry

Formula: The 2 × 2 determinant

$\det \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=ad-bc$

Rows are split by a semicolon. Its absolute value is the area of the image of the unit square.

Example: A 3 × 3 by cofactors

Find $\det \left[\begin{array}{ccc}3& 0& 2\\ 1& 4& 0\\ 2& 1& 5\end{array}\right]$.

1. Expand along row $1$: the $0$ term vanishes
2. First $3\times (4\times 5-0\times 1)=60$, last $2\times (1\times 1-4\times 2)=-14$
3. Total: $60-14=46$

Idea: Row operations

Simplify first, knowing what each move does to the determinant.

• Swap $2$ rows: the sign flips
• Scale a row by $c$: the determinant scales by $c$
• Add a multiple of one row to another: no change

Quick check: Eliminate, then multiply

Find $\det \left[\begin{array}{ccc}2& 4& 2\\ 1& 5& 3\\ 3& 6& 7\end{array}\right]$.

Formula: Stretch factors multiply

$\det (AB)=\det (A)\det (B)$

So det(A⁻¹) = 1/det(A), and det(cA) = cⁿ det(A) for an n × n matrix.

Formula: The 2 × 2 inverse

$A^{-1}=\left(\frac{1}{ad-bc}\right)\times \left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

Swap the diagonal, change the sign of the other entries, divide by the determinant.

Example: One entry of an inverse

For $\left[\begin{array}{cc}5& 2\\ 3& 2\end{array}\right]$, what is the entry in row $2$, column $1$ of the inverse?

1. Determinant: $5\times 2-2\times 3=4$
2. That entry is minus $c$ over the determinant: $-\frac{3}{4}$

Common mistake: The indices swap

Using the cofactor of $(i,j)$ for entry $(i,j)$ of the inverse. For $\left[\begin{array}{ccc}3& 0& 2\\ 1& 4& 0\\ 2& 1& 5\end{array}\right]$ that gives $-\frac{5}{46}$, not $\frac{1}{23}$.

Instead Entry $(i,j)$ of $A^{-1}$ is $\frac{C_{ji}}{\det (A)}$. The swap matters whenever the matrix is not symmetric.

Idea: Under time pressure

Look for zeros before you choose a row to expand.

• No helpful row: use row operations, tracking swaps and scalings
• Multiply a full inverse back by $A$ and check you get $I$
• For a single entry, one cofactor is enough